We also need a way of measuring differences in causal structure (like a “metric” on how causally different spacetimes are).  The most venerable existing proposal has a number of problems.  In particular, they only work for spacetimes defined on the same manifold.

A newer proposal due to Bombelli: Compare isometry classes of spacetimes according to the probability of getting the causal set if n points are selected at random from one of the spacetimes.  As a calculational problem, this is extremely hard.  Moreover, it only works for finite spacetimes.  There hasn’t been much progress with this problem.

Another proposal depends on the notion of Hausdorff distance between two subsets of a metric space.  Gromov generalized this to a distance between metric spaces, corresponding to the minimum possible Hausdorff distance when the two spaces are embedded in any larger space.  But can this be extended to Lorentzian manifolds?

A Lorentzian definition of Gromov-Hausdorff distance is proposed, but it is not clear how good a job it will do of providing good approximate agreement on observables for “nearby” spacetimes.